A new approach to the L p -theory of + b , and its applications - - PowerPoint PPT Presentation

A new approach to the Lp-theory of −∆ + b · ∇, and its applications to Feller processes with general drifts.

Damir Kinzebulatov (McGill and the CRM) www.math.toronto.edu/dkinz October 2015

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The key component of many models of Math. Physics: Brownian motion

A trajectory of a three-dimensional Brownian motion

Brownian motion is modeled by Wiener process Wt (where t 0, W0 = 0)

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The key component of many models of Math. Physics: Brownian motion

A trajectory of a three-dimensional Brownian motion

Brownian motion is modeled by Wiener process Wt (where t 0, W0 = 0) The simplest diffusion processes

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Brownian motion perturbed by a (singular) drift b : Rd → Rd?

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Brownian motion perturbed by a (singular) drift b : Rd → Rd? Varadhan, Strook, Albeverio, Krylov, Carlen and many others . . .

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Brownian motion perturbed by a (singular) drift b : Rd → Rd? Varadhan, Strook, Albeverio, Krylov, Carlen and many others . . . The long search for the cricial singularities of the drift b . . .

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A bridge between Probability and Analysis Probability Analysis Wt ← → −∆

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A bridge between Probability and Analysis Precisely, given a realization Ws ∈ Rd, s < t, we have P[Wt ∈ A]

• Probability

= e(t−s)∆1A

• Ws
• Analysis

i.e. to find the probability we need to solve the heat equation

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A bridge between Probability and Analysis Precisely, given a realization Ws ∈ Rd, s < t, we have P[Wt ∈ A]

• Probability

= e(t−s)∆1A

• Ws
• Analysis

i.e. to find the probability we need to solve the heat equation ⇒ Analytic methods in Probability

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A bridge between Probability and Analysis By analogy: let Xt be a Brownian motion perturbed by drift b : Rd → Rd Then we must have P[Xt ∈ A] =

• e−(t−s)(−∆+b·∇)1A
• (Xs),

s < t

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A bridge between Probability and Analysis By analogy: let Xt be a Brownian motion perturbed by drift b : Rd → Rd Then we must have P[Xt ∈ A] =

• e−(t−s)(−∆+b·∇)1A
• (Xs),

s < t e.g. take b ≡ 0 ⇒ Xt = Wt

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A bridge between Probability and Analysis By analogy: let Xt be a Brownian motion perturbed by drift b : Rd → Rd Then we must have P[Xt ∈ A] =

• e−(t−s)(−∆+b·∇)1A
• (Xs),

s < t e.g. take b ≡ 0 ⇒ Xt = Wt We can solve the heat equation for fairly singular b’s. But . . .

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A bridge between Probability and Analysis . . . will the solutions of the heat equation determine a diffusion process?

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A bridge between Probability and Analysis . . . will the solutions of the heat equation determine a diffusion process? The research program started in 1980s, closely tied to the progress in PDEs, and continuing within emerging areas of Probability (SPDEs). . .

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What singularities of the drift are admissible?

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What singularities of the drift are admissible? d 3

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The best possible result in terms of Lp-spaces Denote Lp = Lp(Rd) Stampacchia . . . Krylov, R¨

• ckner, Stannat and many others

Lp + L∞ (p > d) Ld + L∞

• ✶

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Critical drifts? Example: A vector field having critical singularity b(x) := x|x|−2, x ∈ R3 (clearly, b ∈ Ld + L∞)

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Critical drifts? Example: A vector field having critical singularity b(x) := x|x|−2, x ∈ R3 (clearly, b ∈ Ld + L∞) There is a diffusion process Xt with drift b. In fact, a weak solution of dXt = x|x|−2dt + dWt, X0 = 0

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Critical drifts? Example: A vector field having critical singularity b(x) := x|x|−2, x ∈ R3 (clearly, b ∈ Ld + L∞) There is a diffusion process Xt with drift b. In fact, a weak solution of dXt = x|x|−2dt + dWt, X0 = 0 Replace x|x|−2 with (1 + ε)x|x|−2 and the process will cease to exist

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In other words, singularities of b are critical if they are sensitive to multiplication by constants

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In other words, singularities of b are critical if they are sensitive to multiplication by constants The singularities of a b ∈ Ld are sub-critical

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The classes of critical vector fields previously studied in the literature

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Critical point singularities of the drift The class of form-bounded vector fields1 Fδ :=

• b ∈ L2

loc : lim λ↑∞

• |b|(λ − ∆)− 1

2

• L2→L2

√ δ

• Example: b(x) =

√ δ d−2

2 x|x|−2

(Hardy inequality)

1In relatively elementary terms: Kerman-Sawyer, Chang-Wilson-Wolff 13 / 34

Critical point singularities of the drift The class of form-bounded vector fields1 Fδ :=

• b ∈ L2

loc : lim λ↑∞

• |b|(λ − ∆)− 1

2

• L2→L2

√ δ

• Example: b(x) =

√ δ d−2

2 x|x|−2

(Hardy inequality) Fδ is ‘responsible’ for dissipativity of −∆ + b · ∇ in Lp ⇒ a diffusion via a Moser-type iterative procedure of Kovalenko-Semenov

1In relatively elementary terms: Kerman-Sawyer, Chang-Wilson-Wolff 13 / 34

Critical hypersurface singularities of the drift The Kato class of vector fields Kd+1

δ

:=

• b ∈ L1

loc : lim λ↑∞

• |b|(λ − ∆)− 1

2

• L1→L1 δ
• 2Yu. Semenov, Q. S. Zhang, B. Davies . . . earlier, J. Nash, . . .

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Critical hypersurface singularities of the drift The Kato class of vector fields Kd+1

δ

:=

• b ∈ L1

loc : lim λ↑∞

• |b|(λ − ∆)− 1

2

• L1→L1 δ
• Example:

|b(x)| =

• |x| − 1
• −γ,

γ < 1 is in Kd+1

• 2Yu. Semenov, Q. S. Zhang, B. Davies . . . earlier, J. Nash, . . .

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Critical hypersurface singularities of the drift The Kato class of vector fields Kd+1

δ

:=

• b ∈ L1

loc : lim λ↑∞

• |b|(λ − ∆)− 1

2

• L1→L1 δ
• Example:

|b(x)| =

• |x| − 1
• −γ,

γ < 1 is in Kd+1 Kd+1

δ

is ‘responsible’ for the Gaussian bounds2 for −∆ + b · ∇ ⇒ the Gaussian bounds yield a diffusion

• 2Yu. Semenov, Q. S. Zhang, B. Davies . . . earlier, J. Nash, . . .

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Classes Kd+1

δ

and Fδ play prominent role in Analysis Crucial features: – Defined in terms of the operators that constitute the problem3

3This intuition worked, in particular, in unique continuation for Schr¨

• dinger operators

with form-bounded potentials (Kinzebulatov-Shartser, JFA, 2010)

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Classes Kd+1

δ

and Fδ play prominent role in Analysis Crucial features: – Defined in terms of the operators that constitute the problem3 – Are integral conditions, i.e. the geometry of the singularities is not important (well, . . . ) . . . realizations of random fields (SHE)

3This intuition worked, in particular, in unique continuation for Schr¨

• dinger operators

with form-bounded potentials (Kinzebulatov-Shartser, JFA, 2010)

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Classes Kd+1

δ

and Fδ play prominent role in Analysis Crucial features: – Defined in terms of the operators that constitute the problem3 – Are integral conditions, i.e. the geometry of the singularities is not important (well, . . . ) . . . realizations of random fields (SHE) – What really matters is the relative bound δ > 0 (as in δx|x|−2; has to be small, so that b · ∇ ≤ −∆). For instance, Ld ⊂ F0 :=

δ>0 Fδ.

3This intuition worked, in particular, in unique continuation for Schr¨

• dinger operators

with form-bounded potentials (Kinzebulatov-Shartser, JFA, 2010)

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Classes Kd+1

δ

and Fδ play prominent role in Analysis Crucial features: – Defined in terms of the operators that constitute the problem3 – Are integral conditions, i.e. the geometry of the singularities is not important (well, . . . ) . . . realizations of random fields (SHE) – What really matters is the relative bound δ > 0 (as in δx|x|−2; has to be small, so that b · ∇ ≤ −∆). For instance, Ld ⊂ F0 :=

δ>0 Fδ.

– Are L1, L2-conditions (e.g. Kato class of measure-valued drifts: Bass-Chen, Kim-Song), cf. “b ∈ Ld”

3This intuition worked, in particular, in unique continuation for Schr¨

• dinger operators

with form-bounded potentials (Kinzebulatov-Shartser, JFA, 2010)

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The state of affairs (not so long ago) Lp + L∞ (p > d) Ld + L∞

• ✒

✒ ✒ ✒ ✒ ✒ ✒ ✒ ✒

Ld + L∞ Ld,∞ + L∞

• ✒

✒ ✒ ✒ ✒ ✒ ✒ ✒

Ld,∞ + L∞ Fδ

• ✒

✒ ✒ ✒ ✒ ✒ ✒ ✒

Lp + L∞ (p > d) Kd+1

δ ✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱ ✶

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The search for ‘the right’ class of critical drifts b The next step: b := b1 + b2, b1 ∈ Kd+1

δ

, b2 ∈ Fδ (i.e. b combines critical point and critical hypersurface singularities)

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The search for ‘the right’ class of critical drifts b The next step: b := b1 + b2, b1 ∈ Kd+1

δ

, b2 ∈ Fδ (i.e. b combines critical point and critical hypersurface singularities) The main obstacle: “b ∈ Fδ” destroys Gaussian bounds,

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The search for ‘the right’ class of critical drifts b The next step: b := b1 + b2, b1 ∈ Kd+1

δ

, b2 ∈ Fδ (i.e. b combines critical point and critical hypersurface singularities) The main obstacle: “b ∈ Fδ” destroys Gaussian bounds, and “b ∈ Kd+1

δ

” destroys Lp-dissipativity (crucial for the existing proofs)

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To summarize (so far)

• 1. The two prominent classes of singular vector fields Kd+1

δ

, Fδ are responsible for two fundamental properties of −∆ + b · ∇: “Gaussian bounds”, “dissipativity” (both imply that −∆ + b · ∇ generates a diffusion)

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To summarize (so far)

• 1. The two prominent classes of singular vector fields Kd+1

δ

, Fδ are responsible for two fundamental properties of −∆ + b · ∇: “Gaussian bounds”, “dissipativity” (both imply that −∆ + b · ∇ generates a diffusion)

• 2. It is clear that neither Kd+1

δ

nor Fδ is responsible for the property “to generate a diffusion”

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Part II: “A new hope”

arXiv:1502.07286, arxiv:1508:059834

4Or www.math.toronto.edu/dkinz 19 / 34

From an analyst perspective −∆ + b · ∇ generates a diffusion if it generates a strongly continuous semigroup in the Banach space C∞ := {f ∈ C(Rd) : f vanishes at ∞}

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From an analyst perspective −∆ + b · ∇ generates a diffusion if it generates a strongly continuous semigroup in the Banach space C∞ := {f ∈ C(Rd) : f vanishes at ∞} In other words, we solve the Cauchy problem ∂tu − ∆u + b · ∇u = 0, u(0, ·) = f(·) ∈ C∞ in C∞, i.e. we must have strong continuity: lim

t↓0 u(t, ·) = f(·)

in C∞

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A theorem in Probability Strong continuity property in C∞ ⇒ the fundamental solution of −∆ + b · ∇ is the transition (sub-) probability function of a diffusion . . . a bridge between Probability and Analysis

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“A new hope” The class of weakly form-bounded vector fields F

1 2

δ := {b ∈ L1 loc :

• |b|

1 2 (λ − ∆)− 1 4

• 2→2 δ},

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Weakly form-bounded vector fields Proposition: Fδ1 + Kd+1

δ2

F

1 2

δ ,

δ := δ1 + δ2 Proof (easy): interpolation, Heinz-Kato inequality

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Weakly form-bounded vector fields Proposition: Fδ1 + Kd+1

δ2

F

1 2

δ ,

δ := δ1 + δ2 Proof (easy): interpolation, Heinz-Kato inequality Corollary: F

1 2

δ allows to combine critical point and critical hypersurface singularities

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Weakly form-bounded vector fields The state of affairs as of today

Lp + L∞ (p > d) Ld + L∞

• ✒

✒ ✒ ✒ ✒ ✒ ✒ ✒ ✒

Ld + L∞ Ld,∞ + L∞

• ✒

✒ ✒ ✒ ✒ ✒ ✒ ✒

Ld,∞ + L∞ Fδ

• ✒

✒ ✒ ✒ ✒ ✒ ✒ ✒

Lp + L∞ (p > d) Kd+1

δ ✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱

Fδ F

1 2

δ ❉❉❉❉❉❉❉❉❉❉❉❉

Kd+1

δ

F

1 2

δ

• ③

③ ③ ③ ③ ③ ③ ③ ③ ③ ③

✶

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Weakly form-bounded vector fields b ∈ F

1 2

δ

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Weakly form-bounded vector fields b ∈ F

1 2

δ

We need: C∞-regularity theory of −∆ + b · ∇, b ∈ F

1 2

δ

L2-regularity theory of −∆ + b · ∇, b ∈ F

1 2

δ (JFA, Semenov, 2006)

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Weakly form-bounded vector fields b ∈ F

1 2

δ

We need: C∞-regularity theory of −∆ + b · ∇, b ∈ F

1 2

δ

L2-regularity theory of −∆ + b · ∇, b ∈ F

1 2

δ (JFA, Semenov, 2006)

Even in L2: KLMN theorem doesnt’t apply

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C0-semigroups (Kato, Yosida . . . ) Let b ∈ F

1 2

δ

We need: an operator realization Λ(b) of −∆ + b · ∇ generating a (positivity preserving, contraction) C0-semigroup Tt ∈ B(C∞), i.e. (1) Tt+s = TtTs, T0 = 1 (2) Ttf

s

→ Ts in C∞ as t → s, s 0. (3) d

dtTtf = Λ(b)Ttf

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C0-semigroups (Kato, Yosida . . . ) Let b ∈ F

1 2

δ

We need: an operator realization Λ(b) of −∆ + b · ∇ generating a (positivity preserving, contraction) C0-semigroup Tt ∈ B(C∞), i.e. (1) Tt+s = TtTs, T0 = 1 (2) Ttf

s

→ Ts in C∞ as t → s, s 0. (3) d

dtTtf = Λ(b)Ttf

Precise meaning of ‘generating’: Λ(b)f := lim

t↓0

Ttf − f t , f ∈ C∞

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C0-semigroups (Kato, Yosida . . . ) Let b ∈ F

1 2

δ

We need: an operator realization Λ(b) of −∆ + b · ∇ generating a (positivity preserving, contraction) C0-semigroup Tt ∈ B(C∞), i.e. (1) Tt+s = TtTs, T0 = 1 (2) Ttf

s

→ Ts in C∞ as t → s, s 0. (3) d

dtTtf = Λ(b)Ttf

Precise meaning of ‘generating’: Λ(b)f := lim

t↓0

Ttf − f t , f ∈ C∞ Denote e−tΛ(b) := Tt

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We will ‘design’ the resolvent (λ + Λ(b))−1 ∈ B(C∞), λ > λ0 > 0

• f the required diffusion generator Λ(b)

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A new approach: ‘designing the resolvent’ Our starting object: an operator-valued function on Lp, p is in a bounded open interval depending on the relative bound δ, Θp(λ, b) := (λ − ∆)−1 − (λ − ∆)− 1

2 Qp(1 + Tp)−1Gp,

where Qp = (λ − ∆)− 1

2 |b| 1 p′ ,

Tp = b

1 p · ∇(λ − ∆)−1|b| 1 p′ ,

Gp = b

1 p · ∇(λ − ∆)−1,

b

1 p := b|b| 1 p −1 28 / 34

Formally, Θp(λ, b) =

∞

• k=0

(−1)k(λ − ∆)−1 b · ∇(λ − ∆)−1 . . . b · ∇(λ − ∆)−1

• k times

where the RHS is the Neumann series for (λ + Λ(b))−1 So, Θp(λ, b) is ‘a candidate’ for the resolvent (λ + Λ(b))−1!

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A new approach: ‘designing the resolvent’ Our starting object: an operator-valued function on Lp Θp(λ, b) := (λ − ∆)−1 − (λ − ∆)− 1

2 Qp(1 + Tp)−1Gp

Proposition: If b ∈ F

1 2

δ , then Qp, Tp, Gp ∈ B(Lp)

Proof: Using Lp-inequalitites between (λ − ∆)

1 2 and ‘potential’ |b|

(Liskevich-Semenov, 1996)5

5Note: Kd+1 δ

, Fδ reduce everything to −∆ + b2

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A new approach: ‘designing the resolvent’ Our operator-valued function Θp(λ, b) := (λ − ∆)−1 − (λ − ∆)− 1

2 Qp(1 + Tp)−1Gp 31 / 34

A new approach: ‘designing the resolvent’ Our operator-valued function Θp(λ, b) := (λ − ∆)−1 − (λ − ∆)− 1

2 Qp(1 + Tp)−1Gp

The key insight: If the relative bound δ > 0 (in b ∈ F

1 2

δ ) is small, we can select p > d

31 / 34

A new approach: ‘designing the resolvent’ Our operator-valued function Θp(λ, b) := (λ − ∆)−1 − (λ − ∆)− 1

2 Qp(1 + Tp)−1Gp

The key insight: If the relative bound δ > 0 (in b ∈ F

1 2

δ ) is small, we can select p > d

Then by the Sobolev embedding theorem, (λ − ∆)− 1

2 will map Lp to C∞! 31 / 34

A new approach: ‘designing the resolvent’ Our operator-valued function Θp(λ, b) := (λ − ∆)−1 − (λ − ∆)− 1

2 Qp(1 + Tp)−1Gp

The key insight: If the relative bound δ > 0 (in b ∈ F

1 2

δ ) is small, we can select p > d

Then by the Sobolev embedding theorem, (λ − ∆)− 1

2 will map Lp to C∞!

So, Θp(λ, b)Lp ⊂ C∞

31 / 34

A new approach: ‘designing the resolvent’ Now, to prove that Θp(λ, bn)f

s

→ Θp(λ, b)f in C∞, f ∈ C∞

0 ,

where bn are bounded (smooth) approximations of b, we only need to work in Lp, p > d, a space having much weaker topology

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A new approach: ‘designing the resolvent’ Now, to prove that Θp(λ, bn)f

s

→ Θp(λ, b)f in C∞, f ∈ C∞

0 ,

where bn are bounded (smooth) approximations of b, we only need to work in Lp, p > d, a space having much weaker topology ⇒ the gain in the admissible singularities of the drift

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A new approach: ‘designing the resolvent’ We had to ‘dive in’ into the Lp-theory of −∆ + b · ∇

6Bass-Chen [Ann. Prob. 2003], Chen-Kin-Song [Ann. Prob. 2012] 33 / 34

A new approach: ‘designing the resolvent’ We had to ‘dive in’ into the Lp-theory of −∆ + b · ∇ If we stay6 in C∞ ⇒ b ∈ Kd+1

δ

Note: Kd+1 ensures continuity of ∇etΛ(b)

6Bass-Chen [Ann. Prob. 2003], Chen-Kin-Song [Ann. Prob. 2012] 33 / 34

Concluding remarks This method:

• 1. Also provides a detailed Lp-regularity of −∆ + b · ∇, e.g. characterizes

smoothness of the domain of the generator in terms of δ > 0 (in “b ∈ F

1 2

δ ”)

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Concluding remarks This method:

• 1. Also provides a detailed Lp-regularity of −∆ + b · ∇, e.g. characterizes

smoothness of the domain of the generator in terms of δ > 0 (in “b ∈ F

1 2

δ ”)

• 2. Depends on the fact that −∆ and ∇ commute

34 / 34

Concluding remarks This method:

• 1. Also provides a detailed Lp-regularity of −∆ + b · ∇, e.g. characterizes

smoothness of the domain of the generator in terms of δ > 0 (in “b ∈ F

1 2

δ ”)

• 2. Depends on the fact that −∆ and ∇ commute ⇒ extension to

non-local operators (−∆)

α 2 + b · ∇ (Levy processes . . . ) 34 / 34

Concluding remarks This method:

• 1. Also provides a detailed Lp-regularity of −∆ + b · ∇, e.g. characterizes

smoothness of the domain of the generator in terms of δ > 0 (in “b ∈ F

1 2

δ ”)

• 2. Depends on the fact that −∆ and ∇ commute ⇒ extension to

non-local operators (−∆)

α 2 + b · ∇ (Levy processes . . . )

• 3. b ∈ F

1 2

δ (an L1-condition) can be a measure, e.g. Brownian motion

drifting upward when penetrating certain fractal-like sets (using a variant

• f the Kato-Ponce inequality by Grafakos-Oh, CPDE, 2014)

34 / 34

Concluding remarks This method:

• 1. Also provides a detailed Lp-regularity of −∆ + b · ∇, e.g. characterizes

smoothness of the domain of the generator in terms of δ > 0 (in “b ∈ F

1 2

δ ”)

• 2. Depends on the fact that −∆ and ∇ commute ⇒ extension to

non-local operators (−∆)

α 2 + b · ∇ (Levy processes . . . )

• 3. b ∈ F

1 2

δ (an L1-condition) can be a measure, e.g. Brownian motion

drifting upward when penetrating certain fractal-like sets (using a variant

• f the Kato-Ponce inequality by Grafakos-Oh, CPDE, 2014)
• 4. . . .

34 / 34

35 / 34

fp,∞ :=

• supt>0 tpµ
• |f(x)| > t

1

p 36 / 34

A new approach: ‘designing the resolvent’ Since Θp(λ, bn)f

s

→ Θp(λ, b)f in C∞, f ∈ C∞

0 ,

Θp(λ, bn)L∞→L∞ λ−1

• ‘external’ fact

⇒ Θp(λ, b)fL∞ λ−1fL∞, so we have a well defined the ‘true candidate’ for the resolvent: Θ(λ, b) :=

• Θp(λ, b)|Lp∩C∞

cl

C∞ ∈ B(C∞)

37 / 34

A new approach: ‘designing the resolvent’ Θ(λ, b) satisfies (same argument with the Sobolev embedding theorem) λΘ(λ, b)

s

→ 1 in C∞ as λ ↑ ∞ ⇒ a pseudoresolvent Θ(λ, b) is the resolvent of a densely defined operator

38 / 34

A new approach: ‘designing the resolvent’ Θ(λ, b) satisfies (same argument with the Sobolev embedding theorem) λΘ(λ, b)

s

→ 1 in C∞ as λ ↑ ∞ ⇒ a pseudoresolvent Θ(λ, b) is the resolvent of a densely defined operator Now, λΘ(λ, b)L∞→L∞ 1 (proved in the last slide) ⇒ we can define (λ + Λ(b))−1 := Θ(λ, b)

38 / 34